´ CONFORTI-CORNUEJOLS CONJECTURE VIA COMMUTATIVE ALGEBRA ` HA ` AND SUSAN MOREY HUY TAI Abstract. Using the correspondence between clutters and square-free monomial ideals, we investigate an algebraic approach to a conjecture of Conforti and Cornu´ejols which states that a clutter has the max-flow min-cut property if and only if it has the packing property. If a “minimal” counterexample C to this conjecture existed then powers of the corresponding monomial ideal I must have embedded primes. We show that the least power t such that I t has embedded primes is exactly t = β1 (C) + 1, where β1 (C) is the matching number of C. Our results also show that if existed, such a clutter C cannot be unmixed.
1. Introduction There is a combinatorial realization of a square-free monomial ideal that can be manifested in a variety of ways, depending on the reader’s background. Since the primary motivation of this paper is a conjecture from combinatorial optimization, we will use clutter language to state our problem and results. A clutter C consists of a finite set of points V (C) = {x1 , . . . , xd } and a family E(C) of nonempty subsets of V (C) with no non-trivial containments among them (i.e., if E1 and E2 are distinct elements in E(C) then E1 6⊆ E2 ). The elements of V (C) are called the vertices and the elements of E(C) are called the edges of C. Clutters are also known as Sperner families or simple hypergraphs. Let k be a field. By identifying the points in V (C) with the variables in a polynomial ring R = k[x1 , . . . , xd ], the natural one-to-one correspondence between square-free monomial ideals in R and clutters on {x1 , . . . , xn } is given by Y C ↔ I(C) = hxE = x | E ∈ E(C)i. x∈E
The ideal I(C) is referred to as the edge ideal of C. This is the same as the construction of edge ideals of hypergraphs (cf. [12, 6]). In [2], Conforti and Cornu´ejols made the following conjecture (see Section 2 for the relevant definitions). Conjecture 1.1 (Conforti-Cornu´ejols). A clutter C has the max-flow min-cut (MFMC) property if and only if C has the packing property. The first author is partially supported by BOR Grant LEQSF(2007-10)-RD-A-30 and Tulane Research Enhancement Fund. 1
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In 2001, Cornu´ejols included Conjecture 1.1 as one of 18 conjectures for which he offered a prize (see [3, Conjecture 1.6]). It is well known (see also Section 2) that the MFMC property implies the packing property, so the point of Conjecture 1.1 is the other implication. In [5, Corollary 3.14] and [7, Corollary 1.6], it was shown that C satisfies MFMC if and only if the corresponding edge ideal I(C) is normally torsionfree, that is, I(C)t = I(C)(t) for all t ≥ 1. This allows Conjecture 1.1 to be restated (cf. [4, Conjecture 4.18]) as: if C has the packing property, then I(C)t = I(C)(t) for all t ≥ 1 (or equivalently, I(C)t has no embedded primes for all t ≥ 1). If C is a graph in the classical sense, i.e., I(C) is generated in degree two, then the packing property is equivalent to C being bipartite. In [11] it was shown that a graph is bipartite if and only if its edge ideal is normally torsion-free. Thus, Conjecture 1.1 has been verified for the case of a graph. The goal of this paper is to examine algebraic properties of I(C) when C is a “minimal” possible counterexample (if existed) to Conjecture 1.1. Our work can be viewed as the first step toward obtaining an algebraic solution to Conjecture 1.1. If such a C existed then I(C) was not normally torsion-free, i.e., there existed a power I(C)t that has embedded primes. Thus, our focus is in investigating associated and embedded primes of powers of a square-free monomial ideals. More precisely, we study the associated and embedded primes of I(C)t in the case when every minor (see Section 2 for the definition) of I(C) is normally torsion-free. Our first main result, Theorem 3.9, shows that if, in addition to satisfying packing property, C is unmixed, then I(C) is normally torsion-free. As a consequence (Corollary 3.10), a minimal counterexample, if existed, to Conjecture 1.1 cannot be an unmixed clutter. Our next results (Corollary 3.6 and Theorem 4.6) give a sharp lower bound of the power t for which I(C)t has embedded primes when I(C) is not normally torsion-free. We show that in this case, I(C)t has no embedded primes for all t ≤ β1 (C), and I(C)β1 (C)+1 must have embedded primes, where β1 (C) is the matching number of C. Our method in proving Theorem 3.9 and Corollary 3.6 is to use induction on the power I(C)t of I(C) and the number of vertices in C. More specifically, we relate the set of associated primes of the tth power I(C)t to that of the tth power of minors of I(C) and that of mth powers I(C)m for m < t. To do this, we pass to a colon ideal I(C)t : M , where M is a product of distinct variables in R, and then relate this with smaller powers of I(C) and powers of minors of I(C). Our method in proving Theorem 4.6 is to use polarization. In particular, we develop a correspondence (which is not necessary one-to-one) between the associated primes of I(C)t to that of its polarization.
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2. Background and Definitions A clutter can also be viewed as a simple hypergraph. For this reason, we shall introduce terminology from both clutter and hypergraph languages, and sometimes use these terminology interchangeably. Throughout the paper, C will denote a clutter over d vertices {x1 , . . . , xd } and R = k[x1 , . . . , xd ] will be the corresponding polynomial ring. A vertex x ∈ V (C) is called an isolated vertex if {x} ∈ E(C). By definition, if x is an isolated vertex of C then {x} is the only edge in C that contains x. A clutter is uniform if all of its edges have the same cardinality. A transversal (or vertex cover) of C is a set of vertices that has nonempty intersection with all of the edges. We will primarily be interested in minimal transversals (or minimal vertex covers), where minimality is with respect to inclusion. It is easy to see that there is a one-to-one correspondence between minimal transversals of C and minimal primes of I(C). Since I(C) is a monomial ideal, all minimal primes are monomial ideals, that is, they are generated by subsets of the variables. The minimum cardinality of a transversal of C will be denoted by α0 (C). Note that by the above correspondence, α0 (C) is also the height of I(C). A matching (or independent set) of C is a set of pairwise disjoint edges. We will refer to generators of a square-free monomial ideal I as being independent if the corresponding edges of the associated clutter are independent; that is, the generators have disjoint support. We will primarily be interested in maximal matchings, where maximality is with respect to inclusion. The maximum cardinality of a matching in C will be denoted by β1 (C). Clearly, α0 (C) ≥ β1 (C) for any clutter C. A clutter C is said to satisfy the K¨onig property if α0 (C) = β1 (C). An ideal is said to satisfy the K¨onig property if its associated clutter satisfies the property. There are two operations commonly used on a clutter C to produce a new, related, clutter on a smaller vertex set. For a vertex x ∈ V (C), the deletion C \ x is formed by removing x from the vertex set and deleting any edge in C that contains x. This has the effect of setting x = 0, or of passing to the ideal (I(C), x)/(x) in the quotient ring R/(x). The contraction C/x is obtained by V (C/x) = V (C) \ {x} and E ∈ E(C/x) if x 6∈ E and either E ∈ E(C) or E ∪ {x} ∈ E(C). This process has the effect of setting x = 1, or of passing to the localization I(C)x in Rx . Any clutter formed by a sequence of deletions and contractions is called a minor of C. The edge ideal of a minor of C is also called a minor of I(C). As observed, minors of an edge ideal can be obtained by taking a sequence of quotients and localizations of the original ideal. A clutter C is said to have the packing property if C and all of its minors satisfy the K¨onig property. We say an ideal has the packing property if its associated clutter has the packing property. On a more algebraic note, we will need to use the minimal primes, associated primes and symbolic powers of ideals. A prime P is minimal over an ideal I if I ⊆ P and there does not exist a prime Q 6= P with I ⊆ Q ( P . A prime P is an associated
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prime of I if there exists an element c in R such that P = (I : c). Note that all minimal primes are also associated primes. An ideal I has a primary decomposition I = q1 ∩ . . . ∩ qt ∩ Q1 ∩ . . . ∩ Qs √ were qi and Qj are primary ideals with qi the minimal primes of I. The primes p Qj are the embedded associated primes of I. Observe that if I = I(C) then minimal primes of I correspond to minimal vertex covers of C. The tth symbolic power of an ideal I, denoted by I (t) , is the intersection of the primary components of I t that correspond to minimal primes of I. An ideal I is called normally torsion-free if I t = I (t) for all t ≥ 1. An ideal I is unmixed if all of its minimal primes have the same height. In the language of combinatorial optimization, if I = I(C), this is equivalent to requiring that the Alexander dual (or transversal) clutter of C is uniform. A clutter C is unmixed if I(C) is unmixed, i.e., if all minimal vertex covers of C has the same cardinality. Suppose I is a square-free monomial ideal minimally generated by (xa1 , xa2 , . . . , xat ) where xv is an abbreviation for xv11 xv22 · · · xvdd . The incidence matrix A of I is the matrix whose ith column is ai . The ideal I, or the clutter C associated to I, satisfies the max-flow min-cut (MFMC) property if for all nonnegative integral vectors w ∈ Zd , both sides of the dual linear programming system min{hw, vi | v ≥ 0, vAT ≥ 1} = max{hy, 1i | y ≥ 0, Ay ≤ w} have integral optimal solution vectors v and y. Here 1 refers to the vector all of whose entries are 1, and h·, ·i is the standard inner product. The packing property can also be restated in terms of the dual linear programming system. An ideal has the packing property if and only if the dual linear programming system as above has integral optimal solutions for all (0, 1, ∞)-vectors w, that is, when entries of w are all 0, 1 or ∞. Thus, it is clear that the MFMC property implies the packing property. An important fact that we shall make use of is that localization preserves associated primes. That is, if P is a prime ideal containing I then P ∈ Ass(R/I t ) if and only if P RP ∈ Ass(RP /(IRP )t ). Note that localizing at P is equivalent to passing to a minor of I, thus the packing property is preserved under localization, as is the unmixed property. This allows us to reduce our problem to investigating when the maximal ideal m = (x1 , . . . , xd ) is an associated or embedded prime of I(C)t . 3. Associated Primes and Unmixed Clutters In this section, we study the set of associated primes of powers a square-free monomial ideal I(C) whose every minor is normally torsion-free. Our primary focus is to determine when I(C) is not normally torsion-free. That is, when a power I(C)t has embedded primes. We show that I(C)t does not have any embedded primes for t ≤ β1 (C). We also show that if, in addition, C is unmixed then I(C) is normally torsion-free.
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Recall that R = k[x1 , . . . , xd ]. We shall start by a folklore result (whose proof is elementary), which we state as a lemma for the ease of referring purpose. Lemma 3.1. Let K be an ideal and let x be an element in R. Then the following sequence is exact: x
0 → R/(K : x) → R/K → R/(K, x) → 0. The next few lemmas exhibit the behavior of associated primes of monomial ideals when passing to subrings or larger rings obtained deleting or adding variables. Note that associated primes of monomial ideals are again monomial, and are generated by subsets of the variables. Lemma 3.2. Let K be a monomial ideal in R. Let x an indeterminate of R such that x does not divide any minimal generator of K. Then there is a one-to-one correspondence between the sets Ass(R/K) and Ass(R/(K, x)) given by P ∈ Ass(R/K) if and only if Q = (P, x) ∈ Ass(R/(K, x)). Proof. Suppose P ∈ Ass(R/K). Then there is a monomial c ∈ R such that P = (K : c). Since x does not divide any minimal generator of K, we may assume that x does not divide c. Clearly, (P, x) ⊆ ((K, x) : c). To see the other inclusion, consider a monomial f ∈ ((K, x) : c). If x|f , then f ∈ (P, x). If x does not divide f , then since x does not divide c, we have that x does not divide f c. Since f c is a monomial and (K, x) : c is a monomial ideal, we have f c ∈ K, and so f ∈ (K : c) = P . Now suppose that Q ∈ Ass(R/(K, x)). Since x ∈ (K, x) ⊆ Q and Q is generated by a subset of the variables, we can write Q = (P, x) for some prime ideal P . Let c ∈ R be a monomial such that with Q = (P, x) = ((K, x) : c). If x|c, then ((K, x) : c) = R 6= P , a contradiction. Thus, x does not divide c. Let y ∈ P be a minimal generator. Then x does not divide y and yc ∈ (K, x). This, and because (K, x) is a monomial ideal, implies that yc ∈ K. Therefore, P ⊆ (K : c). Conversely, let g ∈ (K : c) be a minimal monomial generator of (K : c). Since x does not divide any minimal generator of K, we have that x does not divide g. It then follows, since g ∈ (K : c) ⊆ ((K, x) : c) = Q and x does not divide g, that g ∈ P . Lemma 3.3. Let K be a monomial ideal and let M be a monomial in R. Suppose P ∈ Ass(R/(K : M )). Then P ∈ Ass(R/K). Proof. Since P ∈ Ass(R/(K : M )), there exists a monomial c ∈ R such that P = ((K : M ) : c). Since ((K : M ) : c) = (K : M c), we have that P = (K : M c). Thus, P ∈ Ass(R/K). The next lemma will allow us to concentrate on square-free monomial ideals associated to connected clutters. This will be useful when passing to minors, as the minors of a clutter need not be connected. The result is essentially an extension of the preceding lemma and has been proven elsewhere for special cases (see [11, Corollary 5.6] for the normally torsion-free case and see [1, Lemma 2.1] for the case of the edge ideal of a graph).
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Lemma 3.4. Suppose I is a square-free monomial ideal in S = k[x1 , . . . , xt , y1 , . . . , ys ] such that I = I1 S + I2 S, where I1 ⊆ S1 = k[x1 , . . . , xt ] and I2 ⊆ S2 = k[y1 , . . . , ys ]. Then P ∈ Ass(S/I n ) if and only if P = P1 S + P2 S, where P1 ∈ Ass(S1 /I1n1 ) and P2 ∈ Ass(S2 /I2n2 ) with n1 + n2 = n + 1. Proof. Suppose first that Pi ∈ Ass(Si /Iini ) for i = 1, 2, and P = P1 S + P2 S. Then there exist monomials ci ∈ Si such that Pi = (Iini : ci ), for i = 1, 2. Since Iini is a monomial ideal, Pi is a prime ideal generated by a subset of the variables in Si . Thus, it can be seen that ci ∈ Iini −1 \ Iini for i = 1, 2. Now, if u ∈ P1 then uc1 c2 ∈ I1n1 I2n2 −1 S ⊆ I n . Similarly, if v ∈ P2 then vc1 c2 ∈ I n . Thus, P ⊆ (I n : c1 c2 ). On the other hand, let w ∈ S be a monomial such that wc1 c2 ∈ I n . Since the variable sets for S1 and S2 are disjoint, we have c1 c2 ∈ I n−1 \ I n . Write w = w1 w2 where w1 ∈ S1 and w2 ∈ S2 . Observe that if wi ci 6∈ Iini for i = 1, 2 then wc1 c2 6∈ I n , a contradiction. Therefore, wi ∈ Pi for some i and so w ∈ P . For the converse, suppose P ∈ Ass(S/I n ). Observe again that P is generated by a subset of the variables in S, and so we can write P = P1 S + P2 S, where P1 = P ∩ S1 and P2 = P ∩ S2 . Also, there exists a monomial c ∈ S such that P = (I n : c). As above, it can be seen that c ∈ I n−1 \ I n . Write c = c1 c2 , where c1 ∈ S1 and c2 ∈ S2 are monomials. Then c1 ∈ I1k and c2 ∈ I2s for some 0 ≤ k, s ≤ n − 1 with k + s = n − 1. Suppose x is a minimal generator of P1 . Then x ∈ P = (I n : c), so xc1 c2 ∈ I n = I k+s+1 . This implies that xc1 ∈ I1k+1 . Therefore, P1 ⊆ (I1k+1 : c1 ). On the other hand, let u be a monomial in (I1k+1 : c1 ). Then uc1 c2 ∈ I1k+1 I2s S = I n . This implies that u ∈ P . It follows that u ∈ P ∩ S1 = P1 . Therefore, P1 = (I1k+1 : c1 ). A similar argument shows that P2 = (I2s+1 : c2 ). The conclusion follows by setting n1 = k + 1 and n2 = s + 1. As observed before, associated primes behave well under localization, and so our problem can be reduced to examining when the maximal ideal m = (x1 , . . . , xd ) is an associated prime of I(C)t . Proposition 3.5. Let I be a square-free monomial ideal such that every minor of I is normally torsion-free. Let y1 , . . . , ys be distinct variables in R, and let m = (x1 , . . . , xd ) be the maximal homogeneous ideal of R. Then m ∈ Ass(R/I t ) if and only Qs t if m ∈ Ass(R/(I : i=1 yi )). Proof. By repeated use of Lemmas 3.2 and 3.4, we may assume that the associated clutter of I does not contain any isolated vertices. That is, we may assume that all minimal generators of I are of degree at least 2. Note also that by the established direction of Conjecture 1.1, our hypothesis implies that every minor of I satisfies the packing property. Q It follows from Lemma 3.3 that if m ∈ Ass(R/(I t : si=1 yi )) then m ∈ Ass(R/I t ). We shall use induction on s to prove the other direction. By Lemma 3.1, we have the following exact sequence: 0 → R/(I t : y1 ) → R/I t → R/(I t , y1 ) → 0.
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It then follows from [8, Theorem 6.3] that (3.1)
Ass(R/I t ) ⊆ Ass(R/(I t : y1 )) ∪ Ass(R/(I t , y1 )).
Let J be the minor of I formed by deleting y1 . That is, the generators of J are obtained from the generators of I by setting y1 = 0. By abusing of notation, we write J t for both the ideal J t in R/(y1 ) and its extension in R. Note that J t ⊆ I t , and the generators of J t are precisely the generators of I t that are not divisible by y1 . Thus, (I t , y1 ) = (J t , y1 ). By the hypothesis, J is normally torsion-free, and so Ass(R/J t ) = Min(R/J). It follows, since J is square-free, that the maximal homogeneous ideal of R/(y1 ) is not an associated prime of J t unless J consists of isolated vertices. Yet, isolated vertices of J are also isolated vertices of I, and so we may assume that J does not have isolated vertices. Also, by Lemma 3.2, we have P ∈ Ass(R/(J t , y1 )) if and only if P = (P1 , y1 ) where P1 ∈ Ass(R/J t ) = Min(R/J). Therefore, if P = (P1 , y1 ) ∈ Ass(R/(I t , y1 ) then P1 is not the maximal ideal in R/(y1 ). That is, m 6∈ Ass(R/(I t , y1 )). It now follows from (3.1) that if m ∈ Ass(R/I t ) then m ∈ Ass(R/(I t : y1 )). Suppose for s − 1, and m ∈ Ass(R/I t ). Let Qs−1 now that the assertion is proved t M = i=1 yi . By induction, m ∈ Ass(R/(I : M )). By Lemma 3.1, we have the exact sequence 0 → R/((I t : M ) : ys ) → R/(I t : M ) → R/((I t : M ), ys ) → 0. By using [8, Theorem 6.3] again, we have (3.2)
Ass(R/(I t : M )) ⊆ Ass(R/((I t : M ) : ys )) ∪ Ass(R/((I t : M ), ys )).
Let K be the minor of I formed by setting ys = 0. We shall first show that ((I t : M ), ys ) = ((K t : M ), ys ). Indeed, consider a monomial f ∈ ((I t : M ), ys ). If ys |f , then f ∈ ((K t : M ), ys ). If ys does not divide f , then f ∈ (I t : M ), and so f M ∈ I t . Observe that ys does not divide M nor f , so ys does not divide f M . Also, the generators of K t are generators of I t that are not divisible by ys . Thus, f M ∈ K t . That is, f ∈ (K t : M ) ⊆ ((K t : M ), ys ). Conversely, consider a monomial g ∈ ((K t : M ), ys ). If ys |g, then g ∈ ((I t : M ), ys ). If ys does not divide g, then g ∈ (K t : M ), i.e. gM ∈ K t ⊆ I t . Thus, g ∈ (I t : M ) ⊆ ((I t : M ), ys ). By Lemma 3.2, P ∈ Ass(R/((K t : M ), ys )) if and only if P = (P1 , ys ) for some P1 ∈ Ass(R/(K t : M )). By Lemma 3.3, Ass(R/(K t : M )) ⊆ Ass(R/K t ). Also, since K is a minor of I, our hypothesis implies that K is normally torsion-free. That is, Ass(R/K t ) = Min(R/K). Thus, by a similar argument as above, we have that t m 6∈ Ass(R/((K t : M ), ys )) = Ass(R/((I : M ), ys )). This and (3.2) imply that Q s t t m ∈ Ass(R/((I : M ) : ys )) = Ass(R/(I : i=1 yi )). The result is proved. As a consequence of Proposition 3.5, we obtain a lower bound for the least power t such that I(C)t has embedded primes. Notice that associated primes localize, so if
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P is an embedded prime of I(C)t that does not contain any other embedded primes, then we can localize at P and reduce to the case that P is the maximal ideal. Corollary 3.6. Let C be a clutter. Assume that every minor of I(C) is normally torsion-free. If m ∈ Ass(R/I(C)t ) then t ≥ β1 (C) + 1. Proof. For simplicity of notation, let β = β1 (C) and I = I(C). By Proposition 3.5, m Q is associated to I t only if m is associated to (I t : di=1 xi ), where the product is taken over all distinct variables in R. Q Q Let {E1 , . . . , Eβ } be a matching in C. Then di=1 xi is divisible by βj=1 xEj ∈ I β , so Qd Q β have (I t : di=1 xi ) = R. Hence, for t ≤ β1 (C), i=1 xi ∈ I . Thus, for t ≤ β1 (C), we Q m is not an associated prime of (I t : di=1 xi ), and so m is not an associated prime of I t. Remark 3.7. We will see later, in Theorem 4.6, that the bound in Corollary 3.6 is sharp. In the rest of this section, we will focus our attention to unmixed clutters. Our next result provides a better control over the colon ideal appearing in Proposition 3.5. Proposition 3.8. Let C be an unmixed clutter satisfying the packing property, and let I be its edge ideal. Let {E1 , . . . , Eβ1 (C) } be a maximal matching in C, and let gi = xEi Q 1 (C) gi ) = for i = 1, . . . , β1 (C). If t > β1 (C) and I t−β1 (C) = I (t−β1 (C)) , then (I t : βi=1 t−β1 (C) I . Q Proof. For simplicity of notation, let β = β1 (C) and M = βi=1 gi . It is easy to see that (I t : M ) ⊇ I t−β . To prove the other inclusion, consider a monomial h ∈ (I t : M ). That is, hM ∈ I t . Then there exist F1 , . . . , Ft ∈ I and L ∈ R such that hM = LF1 · · · Ft . Let P be a minimal prime of I. Since C is unmixed (equivalently, I is unmixed), we have ht P = α0 (C). Since C satisfies the packing property, this implies that ht P = β. Also, P covers each of the gi ’s. Thus, by the pigeonhole principle, P contains precisely one variable from each gi for i = 1, . . . , β. This implies that M ∈ P β \P β+1 . Moreover, P also covers Fi for i = 1, . . . , t, and so hM ∈ P t . Thus, we must have h ∈ P t−β . Now observe that IP is a complete intersection, and that IP = PP . Thus, we have (I r )P = (IP )r = PPr . This is true for any power r. By our hypothesis, I t−β = I (t−β) . That is, I t−β has no embedded primes. It follows that the primary decomposition of I t−β has the form \ I t−β = Q. √
Q∈Min(R/I)
Localizing at a minimal prime P , we get PPt−β = IPt−β = QP , where Q is the primary ideal associated to P in the above decomposition. This implies that Q = P t−β . As
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a consequence, h ∈ Q. This is true for any Q in the primary decomposition of I t−β . Therefore, h ∈ I t−β . Hence, (I t : M ) ⊆ I t−β and the result is proved. We are now ready to state our result toward Conjecture 1.1 for unmixed clutters. Theorem 3.9. Let C be an unmixed clutter satisfying the packing property. Assume that every minor of I(C) is normally torsion-free. Then I(C) is normally torsion-free. Proof. For simplicity of notation, again let β = β1 (C) and I = I(C). Suppose by contradiction that I is not normally torsion-free. That is, there exists t such that I t has embedded primes. We choose t minimal with respect to this property. Suppose P is an embedded prime of I t . Since associated prime localize and all minors of I are normally torsion-free, we may assume that P = m. By Corollary 3.6, we have t > β. Let {E1 , . . . , Eβ } be a maximal matching in C, and let gi = xEi . AfterQa reindexing of the Qs we may also assume that Qβ variables, β x1 , . . . , xs are variables in i=1 gi . That is, i=1 gi = i=1 xi . By Proposition 3.5, Q Q m ∈ Ass(R/I t ) if and only if m ∈ Ass(R/(I t : si=1 xi )) = Ass(R/(I t : βi=1 gi )). Moreover,Qby the choice of t, I t−β = I (t−β) . Thus, it follows from Proposition 3.8 that (I t : βi=1 gi ) = I t−β . Now, also by the choice of t, m 6∈ Ass(R/I t−β ). Therefore, m 6∈ Ass(R/I t ), which is a contradiction. The result is proved. As a direct consequence of Theorem 3.9, we obtain the following result. Corollary 3.10. A minimal counterexample to the Conforti-Cornu´ejols conjecture cannot be unmixed. Remark 3.11. We, in fact, can make Corollary 3.10 stronger. A careful examination of the proof of Proposition 3.8 shows that if there exists a minimal generator g of I(C) such that for each minimal prime P of I(C), only one generator of P divides g, then if t ≥ 2 and I(C)t−1 = I(C)(t−1) , then I(C)t : g = I(C)t−1 . Thus, following a similar line of arguments as in the proof of Theorem 3.9, if all minors of I(C) are normally torsion-free then I(C) is normally torsion-free. Thus, if a minimal counterexample C to Conjecture 1.1 existed, then every minimal generator of I(C) must be an element of P 2 for some minimal prime P of I(C). Example 3.12. Due to our remark above, one might hope that the packing property would imply the existence of a minimal generator g such that g ∈ P \P 2 for all minimal primes P of I. However, while this appears to be true for graphs (* Note to Tai: I think this is true since PP is equivalent to bipartite and I think the double-covers can be used to result in an odd cycle, but I haven’t yet worked out all of the details. If this path turns out to be interesting, I can come back and finish this. *) it need not be true for general square-free monomial ideals. For example, let I be the ideal of k[x1 , . . . , x6 ] generated by I = (x1 x2 x3 , x4 x5 x6 , x1 x2 x4 , x2 x3 x6 , x1 x4 x5 , x3 x5 x6 ). Then I satisfies the packing property, but P1 = (x1 , x3 , x5 ) and P2 = (x2 , x4 , x6 ) are both minimal primes of I, and for each generator g of I, there is an i ∈ {1, 2} such that g ∈ Pi2 .
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4. Polarization and Embedded Associated Primes In this section, we focus on square-free monomial ideals I whose every minor is normally torsion-free but I is not. We show that in this case, I β+1 has embedded primes, where β is the matching number of the clutter associated to I. This further shows that the bound given in Corollary 3.6 is sharp. Throughout the section, I ⊆ R = k[x1 , . . . , xd ] will denote a square-free monomial ideal whose every minor is normally torsion-free. Our method in this section is to use polarization. The process of polarization replaces a power xti by a product of t new variables x(i,1) · · · x(i,t) . We call x(i,j) a shadow of xi . Thus, the polarization of a power I t of I is a square-free monomial ideal in d · t variables. We will use Iet to denote the polarization of I t and use St for the new polynomial ring. Observe that if x(i,j) divides a minimal generator M of Iet , then x(i,k) divides M for all 1 ≤ k ≤ j. The depolarization of an ideal in St is formed by setting x(i,j) = xi for all i, j. Note that the depolarization of Iet is I t . We begin with a lemma showing that a minimal prime of Iet cannot contain more than one variables that are shadows of the same variable in R. This will restrict the class of primes to be considered when dealing with polarization. Lemma 4.1. Let P is a minimal prime of Iet in St , and suppose x(i,j) ∈ P . Then x(i,k) 6∈ P for all k 6= j. Proof. Let Ct be the associated clutter of Iet . Then P is a minimal vertex cover of Ct . Suppose by contradiction that x(i,j) and x(i,k) are both in P and k 6= j. Without loss of generality, assume k < j. Let v be a minimal generator of Iet that is covered by x(i,j) . From our observation above, v is divisible by x(i,l) for all l ≤ j. In particular, v is divisible by x(i,k) . Thus, P \ {x(i,j) } is a vertex cover of Ct . This is a contradiction to the minimality of P . The lemma is proved. Remark 4.2. Observe that every minimal prime of I lifts to a minimal prime of the polarization Iet of I t for every t. Indeed, if (x1 , . . . , xr ) is a minimal prime of I, then (x1 , . . . , xr ) is a minimal prime of I t . This implies that {x(1,1) , . . . x(r,1) } is a vertex cover for the clutter associated to Iet . This cover is necessarily minimal. In other words, (x(1,1) , . . . , x(r,1) ) is a minimal prime of Iet . Our next lemma shows that the embedded primes of I t also lift to associated primes of Iet , and that associated primes of the Iet depolarize to associated primes of I t . This creates a correspondence, which is not usually one-to-one, between associated primes of I t and associated primes of its polarization Iet . Lemma 4.3. Let I be as above, and let t be a positive integer. (1) Let P ∈ Min(R/Iet ), and let p be the depolarization of P . Then p ∈ Ass(R/I t ).
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(2) Let q ∈ Ass(R/I t ). Then, there is at least one prime Q ∈ Ass(R/Iet ) such that the depolarization of Q is q. Proof. (1) By definition, there exists a monomial c in St such that P = (Iet : c). Let Ct be the clutter associated to Iet . Since Iet is square-free, we may assume that c is square-free and c 6∈ P . Thus, we may take c to be the product of variables of St that are not in P . Since there are subsets of {x1 , . . . , xd } that are minimal primes of I, it follows from Remark 4.2 that the set {x(1,1) , x(2,1) , . . . , x(d,1) } cannot be contained in any minimal prime of Iet . In particular, there exists an i such that x(i,1) is not in P . That is, x(i,1) divides c. For a monomial M in St , define M to be the maximal monomial divisor of M with the property that if x(l,j) divides M for some l and j, then x(l,k) divides M for all k ≤ j. It can be seen from the definition that M is the polarization of some monomial in R. Observe that since x(i,1) divides c, c is non-trivial. Let c1 be the depolarization of c. Then ce1 = c divides c. Thus c1 6∈ I t . Suppose P = {x(i1 ,j1 ) , . . . , x(ir ,jr ) }. Consider x(i1 ,j1 ) ∈ P . We have x(i1 ,j1 ) c ∈ Iet , and so there is a monomial generator v1 of I t and an monomial f1 ∈ St such that x(i1 ,j1 ) c = f ve1 , where ve1 is the polarization of v1 in St . Moreover, by the definition of polarization and by the maximality of c, ve1 must divide x(i1 ,j1 ) c. Thus the depolarization of x(i1 ,j1 ) c is in I t . This element is of the form xsi1 c1 for some s. Choose si1 minimal such that s s −1 xi1i1 c1 ∈ I t . Since c1 6∈ I t , si1 > 0. Set c2 = xi1i1 c1 . Then, c2 6∈ I t , xi1 c2 ∈ I t , and ce2 divides c. Consider x(i2 ,j2 ) ∈ P . As above, we have x(i2 ,j2 ) c ∈ Iet , and so x(i2 ,j2 ) c = f2 ve2 for some minimal generator v2 ∈ I t and some f2 ∈ St . By a similar line of arguments as above, ve2 divides x(i2 ,j2 ) c, and the depolarization of xsi2 c2 of x(i2 , j2 )c is in I t . Take s s −1 si2 minimal so that xi2i2 c2 ∈ I t . Again, note that si2 > 0. Set c3 = xi2i2 c2 . Then c3 6∈ I t , xi2 c3 ∈ I t , and ce3 divides c. Notice further that xi1 c3 ∈ I t , since xi1 c2 ∈ I t and c2 divides c3 . By repeating this process, we may assume that cr has been defined so that cr 6∈ I t , xil cr ∈ I t for 1 ≤ l ≤ r, and cer divides c. Thus p ⊆ (I t : cr ). To see that the equality holds, consider a monomial M ∈ (I t : cr ) \ p. Without loss of generality, we may assume M is minimal in the sense that no proper divisor of M satisfies this condition. Since the highest power of each variable appearing in minimal generators of I t is t, the minimality of M implies that the power of each variable appearing in M cr is at g most t. Thus, polarization of M cr makes sense in St , and we have M cr ∈ Iet . Observe g ccer for some M c ∈ St that depolarizes to M . Since cer that M cr can be written as M t cc ∈ Ie . It follows that M c ∈ P . That is, x(i ,j ) divides M c divides c, we have that M k k for some 1 ≤ k ≤ r. Therefore, xik divides M for some 1 ≤ k ≤ r, and so M ∈ p. (2) By definition, there exists a monomial b ∈ R \ q such that q = (I t : b). Suppose xi ∈ q and si ≥ 0 is maximal such that xsi i divides b. Let v be a minimal generator
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of I t and let f be a monomial in R such that xi b = f v. If xi divided f , then b ∈ I t , which is a contradiction. Thus, xsi i +1 divides v, and in particular, si + 1 ≤ t. f et Therefore, the polarization of xi b exists in St , and xf i b = f v ∈ I . By the choice of si , t t x(i,si +1)eb ∈ Ie . Therefore, x(i,si +1) ∈ (Ie : eb). Let Q be the prime ideal of St generated by {x(i,si +1) xi ∈ q}. We have seen that Q ⊆ (Iet : eb) and Q depolarizes to q. We shall show that Q = (Iet : eb). Indeed, consider a monomial z ∈ (Iet : eb). Then, zeb ∈ Iet . That is, zeb is a multiple of a generator of Iet . Since the generators of Iet are the polarizations of the generators of I t , there exists a minimal generator w of I t and h ∈ St such that zeb = hw. e We may assume that (z, h) = 1. Let y be the depolarization of z. Then yb is a multiple of w. Thus, y ∈ (I t : b) = q. Thus xi divides y for some xi ∈ q, and so x(i,j) divides z for some j. Observe that if j ≤ si , then x2 divides zeb and so since the minimal generators of (i,j)
Iet are square-free, x(i,j) divides h, a contradiction to our assumption that (z, h) = 1. Suppose that j ≥ si + 2. Then by the definition of si , x(i,si +1) does not divide ze c. Therefore, x(i,si +1) does not divide hw. e This together with the definition of polarization and the fact that w e is a minimal generator of Iet imply that x(i,l) does not divide w e for l ≥ si + 1. In particular, x(i,j) does not devide w. e Thus, x(i,j) divides h, again a contradiction to our assumption that (z, h) = 1. Thus, j = si + 1, and we have x(i,si +1) divides z, and so z ∈ Q. This is true for any monomial z ∈ (Iet : eb), so Q ⊇ (Iet : eb). The result is proved. Remark 4.4. Notice that in the proof of Lemma 4.3, Q was determined by q and by a fixed b. Thus, Q need not be a unique minimal prime of R/Iet corresponding to q. Example 4.5. Let R = k[x, y, z] and let I = (xy, yz, xz) be the edge ideal of a triangle. Then I 2 = (x2 y 2 , y 2 z 2 , x2 z 2 , xy 2 z, xyz 2 , x2 yz) and Ie2 = (x1 x2 y1 y2 , y1 y2 z1 z2 , x1 x2 z1 z2 , x1 y1 y2 z1 , x1 y1 z1 z2 , x1 x2 y1 z1 ). The associated primes of R/I 2 are {x, y}, {x, z}, {y, z}, {x, y, z} and the associated primes of S/Ie2 are {x1 , y1 }, {x2 , y1 }, {x1 , y2 }, {x1 , z1 }, {x2 , z1 }, {y1 , z1 }, {y2 , z1 }, {x1 , z2 }, {y1 , z2 }, {x2 , y2 , z2 }. Using the correspondence between associated primes of I t and of its polarization, we are now ready to prove our main result in this section. Theorem 4.6. Let I ⊆ R = k[x1 , . . . , xd ] be a square-free monomial ideal such that every minor of I is normally torsion-free. Suppose that the associated clutter C of I does not have the packing property. Then m ∈ Ass(R/I β1 (C)+1 ).
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Proof. For simplicity of notation, let β = β1 (C). By definition, it can be seen that Qd β β+1 . Thus no minimal generator of I β+1 is square-free. This implies i=1 xi ∈ I \ I β+1 . that A = {x(1,2) , x(2,2) , . . . x(d,2) } is a vertex cover of the associated clutter C 0 of Ig We claim that A is a minimal vertex cover of C 0 . Suppose by contradiction that A is not. Then there is a subset B of A that is a minimal vertex cover of C 0 . Let Q be the prime ideal generated by elements in B. By Lemma 4.3, Q depolarizes to an associated prime q of I t . Since every minor of I is normally torsion-free, a localization argument shows that the only possible embedded prime of I t is the maximal homogeneous ideal m of R. This implies that Q depolarizes to a minimal prime of I t . That is, q is a minimal prime of I t , and so q is a minimal prime of I. By reindexing the variables in R if necessary, we may assume that q = (x1 , . . . , xs ). Then C = {x1 , . . . , xs } is a minimal vertex cover of C. By definition, for each xi ∈ C, there exists a monomial generator gi of I such that gi is not covered by C \{xi }. It follows from our hypothesis and the established direction of Conjecture 1.1 that every minor of C has the packing property. Thus, our hypothesis implies that C does not have the K¨onig property. That is, α0 (C) > β1 (C). As observed before, α0 (C) = ht I, so s ≥ α0 (C) ≥ β + 1. Q Observe now that for any j = 1, . . . , s, x2j does not divide M = β+1 i=1 gi . This implies β+1 is not covered by (x f in Ig that the polarization M (1,2) , . . . , x(s,2) ), a contradiction to the fact that B is a vertex cover of C 0 . We have shown that A is a minimal vertex cover of the clutter C 0 associated to β+1 . Let P be the ideal generated by elements in A. Then P is a minimal prime of Ig β+1 . It then follows from Lemma 4.3 that m, which is the depolarization of P , is an Ig associated prime of I β+1 . The result is proved. *** Not sure if we want to include the following! *** Theorem 4.6 is also provides a means for searching for embedded primes through localization. For such a prime P = (I t : c), it is often useful to have a concrete description for the monomial c. Our next results give useful information about such an element, as well as an alternate proof to our bound in Corollary 3.6. Lemma 4.7. Let C be a clutter with edge ideal I. Suppose that every minor of I is normally torsion-free, and m = (I t : c) for some t ≥ 1. Then, x divides c for every variable x in R. In particular, c ∈ I β1 (C) , and so t ≥ β1 (C) + 1. Proof. Suppose there exists a variable x in R that does not divide c. Consider the exact sequence 0 → R/((I t : c) : x) → R/(I t : c) → (R/((I t : c), x) → 0. Since m = (I t : c), we have xc ∈ I t . Thus, the left module of the exact sequence vanishes. Also, since x does not divide c, by a similar argument as in the proof of Proposition 3.5, we have ((I t : c), x) = ((J t : c), x), where J is the minor of I formed by setting x = 0. Our exact sequence now becomes: 0 → R/m → R/((J t : c), x) → 0.
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This implies that m = ((J t : c), x). It then follows that the maximal homogeneous ideal m of R/(x) is an associated prime of J t . However, since I has no isolated vertices, neither does J. Thus, m is an embedded prime of J. This is a contradiction to the hypothesis that J was normally torsion-free. Now let E1 , . . . , Eβ1 (C) be a matching in C. Then since these edges are pairwise Q 1 (C) Ei disjoint and every variable in R divides c, it can be seen that βi=1 x divides β1 (C) t c. Thus, c ∈ I . Moreover, since m 6= R, we must have c 6∈ I . Therefore, t ≥ β1 (C) + 1. Proposition 4.8. Let C be a clutter satisfying the packing property, and let I be its edge ideal. Assume that C does not consist of all Qisolated vertices, every minor of I is normally torsion-free, and m = (I t : c). Then di=1 xi divides c. Moreover, for every minimal prime P of I with ht P = ht I, there exists x ∈ P such that x2 divides c. Proof. The first statement follows from Lemma 4.7. We shall prove the second statement. For simplicity of notation, let α = α0 (C) = ht I and β = β1 (C). It follows from our hypothesis that ht P = α = β. Since C does not consist of only isolated vertices, m is not a minimal prime of I(C). Thus, there exists a variable y in R such that y 6∈ P . Since m = (I t : c), we have that yc = F1 · · · Ft h for some minimal generators Fi ’s of I and h ∈ R. By Lemma 4.7, t ≥ β + 1. Moreover, for any i, Fi ∈ P (since P is a minimal prime of I so P corresponds to a minimal vertex cover of C). Therefore, there exists a variable x ∈ P that divides at least two of the Fi ’s. This implies that x2 divides yc. Now since y 6∈ P , which implies that y 6= x, and so x2 divides c. Notice that for every minimal vertex cover of size α, such an x exists. So unless x is an element of every minimal vertex cover of size α, there must be at least two variables whose squares divide c. 3. We show how to use a generalization of the red/blue arguement to ”work outward” from a minimally non-ntf minor. 4. We give special cases: connected in codim 1 was one we talked about, also n variables any d of which formed an edge; special cycles (eg; cycle of length n, pure generators of degree three, eg: abc, bcd, cde, .... until you cycle back, I’m drawing a blank on the name here). In most of the special cases, we showed what was associated (ex: m is associated when ....). I have a collection of these types of results. I don’t think we had the entire picture at the time, but we might be able to use our results from the proceeding section to clean it up. Example: if before we could show what did occur and where, we might now be able to argue that we had found it all by localizing and showing that m could not have appeared earlier or something. 5. We had very specific examples such as the open n-simplex where we really could prove everything appeared exactly where we wanted.
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References [1] J. Chen, S. Morey, A. Sung The Stable Set of Associated Primes of the Ideal of a Graph, Rocky Mountain J. Math. 32, 71-89, 2002. [2] M. Conforti, G. Cornu´ejols, Clutters that Pack and the Max Flow Min Cut Property: A Conjecture, in The Fourth Bellairs Workshop on Combinatorial Optimization W.R. Pulleyblank and F.B. Shepherd, eds. 1993. [3] G. Cornu´ejols, Combinatorial Optimization: Packing an Covering, CMBS-NSF Regional Conference Series in Applied Mathematics 74, SIAM 2001. [4] I. Gitler, E. Reyes, R.H. Villarreal, Blowup Algebras of Square-free Monomial Ideals and Some Links to Combinatorial Optimization Problems, to appear: Rocky Mountain J. Math. [5] I. Gitler, C. Valencia, R.H. Villarreal, A Note on Rees Algebras and the MFMC Property, to appear: Beitrage Algebra Geom. [6] H.T. H` a and A. Van Tuyl. Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers, J. Algebraic Combinatorics 27 (2008), 215–245. [7] J. Herzog, T. Hibi, N.V. Trung, X. Zheng, Standard Graded Vertex Cover Algebras, Cycles and Leaves, arXiv:math.AC/0606357v1, 2006. [8] H. Matsumura, Commutative Ring Theory, Cambridge Stud. Adv. Math. 8, Canbridge Univ. Press, Cambridge, 1986. [9] K.N. Raghavan, Initial Ideals of Tangent Cones to Schubert Varieties in Orthogonal Grassmannians, preprint, 2007 arXiv:0710.2950v1. [10] A. Schrijver, Combinatorial Oprimization, Polyhedra and Efficiency in: Algorighms and Combinatorics, 24, Springer-Verlag, Berlin, 2003. [11] A. Simis, W.V. Vasconcelos, R.H. Villarreal, On the Ideal Theory of Graphs, J. Algebra 167, 389-416, 1994. [12] R. H. Villarreal. Monomial algebras. Monographs and Textbooks in Pure and Applied Mathematics, 238. Marcel Dekker, Inc., New York, 2001. Tulane University, Department of Mathematics, 6823 St. Charles Avenue, New Orleans, LA 70118 E-mail address:
[email protected] URL: http://www.math.tulane.edu/~tai/ Department of Mathematics, Texas State University, 601 University Drive, San Marcos, TX 78666 E-mail address:
[email protected] URL: http://www.txstate.edu/~sm26/
